Energy harvesting by piezoelectric flags

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Yifan Xia

To cite this version:

Yifan Xia. Energy harvesting by piezoelectric flags. Fluids mechanics [physics.class-ph]. Ecole Poly-technique X, 2015. English. �tel-01571574�

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Laboratoire d’Hydrodynamique, École Polytechnique Unité de Mécanique, ENSTA-ParisTech

Thèse présentée pour obtenir le grade de

Docteur de l’Ecole Polytechnique

Specialité : Mécanique

par

Yifan Xia

Energy harvesting by piezoelectric flags

∗

Soutenue le 6 novembre 2015 devant le jury composé de :

Rapporteurs : _{Benjamin Thiria} - Université Paris Diderot Olivier Thomas - ENSAM Lille

Directeurs : _{Sébastien Michelin - Ecole Polytechnique} Olivier Doaré - ENSTA ParisTech Examinateurs : _{Nigel Peake} - University of Cambridge

Aurélien Babarit - Ecole Centrale de Nantes

∗

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Remerciements

Mes premiers remerciements s’adressent à ma femme, Xiao, que j’aime et qui a toujours su me soutenir pendant les périodes de difficultés et de détresses.

Je remercie mes deux directeurs de thèse, Sébastien Michelin et Olivier Doaré, pour leur encadrement et surtout pour leur patience. Merci de m’avoir guider avec leur grandes disponibilités tout au long de ma thèse. Sans leur enthousiasme et rigueur, cette thèse ne verrait jamais le jour.

Mes remerciements s’adressent également aux membres de jury qui m’ont fait honneur d’être présentés à ma soutenance. Merci à Olivier Thomas et Benjamin Thiria pour leurs rapports et remarques pertinentes sur mon travail. Merci aussi à Nigel Peake et Aurélien Babarit de leur participation à mon jury et de leur questions intéressantes.

Je n’oublierai pas les gens des deux laboratoires où j’ai passé ma thèse. Merci à Jean-Marc Chomaz, et puis à Christophe Clanet pour m’avoir accueilli au LadHyX. Merci à Antoine Chaigne, et puis à Habibou Maitournam pour avoir facilité mon accès à l’UME. Un grand merci à Thérèse, Sandrine, Delphine, et Caroline (que j’ai pu revoir à l’UME après son départ du LadHyX) pour leur gestion impeccable de l’aspect administratif de la vie des doctorants. Je remercie aussi Antoine, Dani, Caroline et Toai pour avoir géré d’une manière efficace le système informatique du LadHyX. Côté l’UME, je remercie Thierry et Nico pour leur disponibilité et patience à m’aider à résoudre de nombreux problèmes que j’ai rencontrés pendant mes expériences.

Merci à Eunok, Chakri, Xavier, Mathieu, Manu, Onofrio, Julien, Tristan, Océane, et tout autres membres du LadHyX d’avoir construit un laboratoire qui possède à la fois une excellence dans la recherche, et une très bonne ambiance que tous ceux qui y ont vécu n’oublieront pas. Merci à Yuan, Julie, Arnaud, Joosung, Corinne, Jean, Romain et Benjamin qui, pendant mes très ponctuels passages Ãă l’UME, m’ont accueilli et m’ont accompagné à l’approche de ma soutenance pour améliorer la qualité de ma présentation. Un merci particulier à Emmanuel et Miguel, avec qui j’ai pu avoir beaucoup d’échanges intéressants, qui d’ailleurs ne sont pas restraints dans le plan de nos recherches.

Enfin, je remercie ma famille, mes parents, qui sont loins mais qui me soutiennent sans faille, et mes amis, sans qui ma vie en France ne serait pas aussi agréable.

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Introduction

1.1 Overview of flow energy harvesting

The development of techniques that extract energy from flowing air and water dates back to the first century A.D., and lasts to our days [Shepherd 1990]. One of the most ancient and well-developed methods of harvesting energy from wind is the windmills that were widely used during the 17th and 18th centuries in Western Eu-rope (Fig.1.1a). Due to the increasing awareness of the scarcity and non-renewable nature of fossil fuels, windmills attract a renewed attention, and modern wind tur-bines (Fig. 1.1b), based on a similar mechanism of ancient windmills, are being rapidly developed in many countries. Wind turbines generates energy through their rotary propeller, and the driving force of the latter’s rotation is not limited to air flow. Novel concepts such as tidal turbines are recently receiving an increasing atten-tion (Fig. 1.1c). Tidal turbines work under the same mechanism as wind turbines, with a major difference being that the rotation of their propeller is driven by tidal waves.

The underlying mechanism of harvesting flow energy using both wind and tidal turbines is the solid bodies’ motion induced by surrounding fluid flows. The interest of using flow-induced motions of solid bodies as means of energy harvesting also gives rise to a renewed interest in a larger thematics of mechanics, i.e. the fluid-solid interactions, as potential energy-harvesting mechanisms. Many recent works focused on canonical flow-induced vibrations, such as the coupled mode flutter of an airfoil [Peng 2009,Zhu 2012,Boragno 2012], or vortex-induced vibrations of a rigid cylinder or a flexible cable [WEB1 ,Bernitsas 2008,Grouthier 2014].

Another canonical and widely studied example of fluid-structure interactions is the flutter instability of flexible plates, or flags. The present work will focus on this phenomenon and explore its potential for energy harvesting.

1.2 Flutter instability

A flexible plate, or a flag, placed in a flow would stay at rest (left, Fig. 1.3) or flap (right, Fig. 1.3) as a result of a competition between its rigidity, which tends to restore the plate to its position of static equilibrium, and the pressure applied by the surrounding flowing fluid, which pushes the plate away from this equilibrium. Another decisive factor for the flutter motion to occur is the flag’s inertia: a flag would not flap if it has no inertia.

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(a) (b)

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Figure 1.1: (a) A medieval wind mill in Saint-Chinian, France, (b) offshore wind turbines of Vattenfall company and (c) artistic view of a tidal turbine by SABELLA (Source: WEB).

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Figure 1.2: (a) Schematic representation of an airfoil and (b) VIVACE, an energy-harvesting device based on vortex-induced vibration [WEB1].

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1.2. Flutter instability 3

Figure 1.3: National flags at rest (left, photo taken during COP10 in Nagoya, Japan, 18–29, Oct. 2010, courtesy to Malaka Rodrigo) and flapping (right, Source: web)

Compared with previously mentioned flow-induced motions, such as the airfoil flutter or the vortex-induced vibration (VIV) of rigid cylinders, which are motions of rigid bodies involving only a limited number (usually 1 or 2) of degrees of freedom, the flexibility of the flag induces large deformations of the structure itself, thus a large number of degrees of freedom. The dynamics of a flag flapping in a flow is influenced not only by the velocity and direction of the incoming flow, but also by the physical properties of the flag, including its dimension and material character-istics. In terms of modelling, due to the large number of degrees of freedom, this problem involves a strong coupling of governing equations of both fluid dynamics and elasticity, hence the difficulty of solving this problem.

The earliest works on this problem are mainly experimental studies [Taneda 1968] with theoretical analysis using the potential flow theory [Wu 1961,

Kornecki 1976]. Recently, with the advent of powerful scientific computing tools and sophisticated experimental techniques, the flutter instability of a flag receives an unprecedented popularity among researchers in the field of both fluid and solid mechanics. One may divide existing works into two main categories: studies of the stability and dynamics of a single flexible plate, and those of several flexible plates in a uniform flow.

1.2.1 Stability and dynamics of a single flag placed in a uniform flow

A great number of researchers have reported studies on stability and dynamics of a single flag placed in a uniform flow [Huang 1995, Eloy 2007, Shelley 2011]. In their experimental studies, Zhang et al. [Zhang 2000] observed the motion of a fil-ament in a 2D flow generated by a soap-film. They observed that depending on the length of the filament, it has two fundamental states: (i) the stretched-straight state when the length is small, and (ii) the flapping state when the length is large. They also observed the evolution of the flapping amplitude and frequency with the filament’s length, and identified a hysteresis phenomenon by increasing and reducing the filament’s length. Watanabe et al. investigated experimentally the paper flutter

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using different materials [Watanabe 2002b], and also developed theoretical models to account for various observations in the experiments [Watanabe 2002a]. In both studies, relations between the flutter speed and the mass ratio, i.e. the ratio between the inertia of the fluid and solid, are identified. Other experimental and theoreti-cal works also investigated the stability [Lemaitre 2005] and post-critical dynamics [Eloy 2012, Virot 2013, Gibbs 2014] of a flag placed in a wind tunnel and offered a vast catalogue of flutter properties of flags made of different materials, such as paper, plastic, fabrics, and metals.

Recently, with the increasing capacity of scientific computing, a large amount of numerical work has been conducted to study the flapping of a flag. The ex-periments of Zhang et al. were reproduced by direct numerical simulation using immersed boundary method (IBM) [Zhu 2002] and arbitrary Lagrangian Eulerian (ALE) [Sawada 2006]. Both numerical methods gave results that are in a good qualitative agreement with the experimental results: (i) stretched-straight state is observed with short flags, (ii) bistability–switching between the stretched-straight state and flapping state–is observed for longer flags.

Despite the benefit that the direct numerical simulation provides a detailed de-scription of the dynamics of both the flow and the structure, it is time-consuming and prohibitive for high Reynolds numbers. As a result, many simplified models are developed to carry out faster simulations. A very popular model for describ-ing the dynamics of a flag is the inextensible Euler-Bernoulli beam model, while a variety of models for the fluid flow, based on the inviscid flow assumption and in-compressibility, are used by different researchers. Alben et al. used a flexible body vortex sheet model to compute the fluid forcing and the flow field around a flap-ping 2D flag [Alben 2008a]. They reported that in addition to the two previously mentioned states: i.e. the stretched-straight and periodic flapping states, a chaotic state, characterised by undefined amplitude and frequency, may appear when the incoming flow velocity is much larger than the critical velocity. Using a unsteady point vortex model, Michelin et al. also identified the existence of this chaotic state [Michelin 2008].

The experimental and numerical techniques for studying a 2D flag in a uniform 2D incoming flow are well developed and have been providing interesting insights of flag’s flapping dynamics. However, the major drawback of these techniques is that they consider flags of an infinite span, which is unrealistic. Many studies therefore also focus on 3D effects on the flapping flag. In their work, Eloy et al. highlighted that the flag’s span has a significant influence on the flag’s stability [Eloy 2007]: with a fixed flag’s length, the onset of flapping takes place at a lower velocity for a flag with larger span. This conclusion is supported by the study of Gibbs et al. [Gibbs 2012], in which experiments are performed and a stability analysis is carried out using the Euler-Bernoulli beam model to describe the flag, and a vortex lattice model [Tang 2007] to account for fluid loading. Another model for the fluid forcing, called Large-Amplitude Elongated-Body Theory (LAEBT), initially developed as to describe fish locomotion [Lighthill 1971,Candelier 2011], is recently adapted to the case of 3D flapping flags [Singh 2012b, Eloy 2012, Michelin 2013]

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1.2. Flutter instability 5 10−1 100 101 102 0 5 10 15 20 25 M∗ U∗ c Stable U nstable

Figure 1.4: Evolution of the critical velocity U_c∗ as a function of M∗. A flag with a given M∗ becomes unstable if the flow velocity exceeds U_c∗ (Figure obtained using the method presented in Chapter3).

by adding drag terms corresponding to the dissipation induced by the lateral flow separation due to the finite flag span. Meanwhile, DNS techniques are also developed for 3D simulations using both IBM [Tian 2012] and ALE [Bourlet 2015].

In many recent studies, a uniform, inviscid, incompressible flow and the Euler-Bernoulli beam model are used to investigate the flag’s flapping in a flow. Using these models, the system is controlled by three dimensionless parameters: the mass ratio M∗, the reduced velocity U∗, and the aspect ratio H∗. These parameters are defined as: M∗= ρ s fL µ , U ∗ _{= U} ∞L r µ B, H ∗₌ H L, (1.1)

where ρs_f and µ are respectively the fluid’s mass per unit surface and the flag’s mass per unit length, L is the length of the flag, U∞ is the incoming flow velocity, B

is the flag’s bending rigidity, and H is the flag’s span. Note that in the 3D case, ρs

f = ρfH, with ρf representing the fluid’s density. Regardless of the various models

used in different works, a widely approved conclusion is that for a flag with a given aspect ratio H∗ (H∗ = ∞ in 2D cases), larger mass ratio M∗ leads to lower critical velocity in terms of U∗, as shown in Fig.1.4.

When U∗ < U_c∗, the flag is stable and stays in the stretched-straight state. Once U∗ > U_c∗, the flag becomes unstable and reaches either a periodic flapping state (Fig. 1.5a, b), or a chaotic flapping state depending on the flow velocity (Fig. 1.5c, d).

Another aspect involved in the studies of a single flag in a uniform flow is the ef-fect of walls. Mainly three configurations are under active investigation: (i) the close presence of one single rigid wall parallel to the flag’s plane [Nuhait 2010,Dessi 2015], (ii) the close presence of two rigid walls parallel to the flag’s plane, thus forming a transverse confinement [Belanger 1995, Guo 2000, Alben 2015], and (iii) the close presence of two rigid walls orthogonal to the flag’s plane, therefore forming a

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span-0 0.5 1 −0.2 0 0.2 x y (c) 60 65 70 −1.6 0 1.6 θ t (d) 0 0.5 1 −0.2 0 0.2 y (a) 40 45 50 −1.6 0 1.6 θ (b)

Figure 1.5: (a, c)Flapping motions and (b, d) of the flag’s trailing edge orientation θ at (a, b) periodic flapping state with M∗= 10, U∗ = 9 and (c, d) chaotic flapping state with M∗ = 10, U∗ = 18.

wise confinement [Doaré 2011b,Doaré 2011c]. These studies showed that the pres-ence of one or two walls in the vicinity of the flag has a destabilising effect, i.e. the confinement reduces the critical velocity. Some studies also reported that the confinement leads to an increase of the flag’s added mass [Belanger 1995,Guo 2000]. Post-critical behaviour of a flag in a transverse confinement is also studied numer-ically by Alben [Alben 2015], who found that while decreasing the channel wall distance from infinity, the flapping amplitude starts by increasing, then decreases because it is limited by the wall. Note that in a wind tunnel test with a flag, both transverse and spanwise confinements may exist depending on the size of the wind tunnel’s test section.

1.2.2 Stability and dynamics of several flags placed in uniform flow

Studying the coupled motion of several flexible bodies placed in a flow is moti-vated by natural phenomena, particularly the fish schooling [Cushing 1968]. Weihs [Weihs 1973] pointed out that in a 2D plane, the optimal positioning of each fish in a school should have a diamond pattern (Fig.1.6) so that the fish that follow others would profit from the thrust induced by the oscillatory motion of their predecessors. The constantly improving techniques for studying a single flag’s flapping are providing new methods to fulfil researchers’ motivation in studying the flapping of multiple flags. Zhang et al. conducted experiments using two filaments placed side by side in a 2D flow based on a soap film [Zhang 2000]. Their results show that under the same incoming flow, two filaments flap in an in-phase pattern (two flags

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1.2. Flutter instability 7

Figure 1.6: Horizontal layer of fish in a school, from above. The diamond pattern is shown with the dashed line (Adapted from [Weihs 1973])

.

have the same vertical displacement) when the distance separating them is small. The flapping becomes out of phase (two flags have the opposite vertical displace-ment) when the two filaments are moved away from each other. The observation of the two flapping patterns, i.e. the out-of-phase one and the in-phase one, is also reproduced by numerical simulations of parallel 2D flags [Zhu 2003,Farnell 2004]. Using the vortex sheet model and the Euler-Bernoulli beam, Alben [Alben 2009b] reported that the phase difference between two side-by-side 2D flags evolves almost monotonically with the distance separating them.

Jia et al. [Jia 2007] performed more thorough experimental investigations of two identical side-by-side filaments placed in a flowing soap film, and studied theoreti-cally their linear stability. They suggest that the coupled dynamics of two filaments is subject to three dimensionless parameters: the mass ratio M∗, the reduced veloc-ity U∗, and the dimensionless form of the separation distance d, defined by:

d = D

L, (1.2)

where D is the dimensional form of the separation distance. According to the vari-ation of these parameters, four flapping modes may be identified: (i) the stretched-straight mode, (ii) in-phase mode, (iii) out-of-phase mode, and (iv) an indefinite mode, i.e. a phase difference switching between 0 and π. Using a double-wake model, Michelin & Llewellyn Smith [Michelin 2009] studied the linear stability of two side-by-side flags and observed that a decreasing d induces to a destabilising ef-fect, thereby lowering the critical velocity U_c∗. Wang et al. [Wang 2010] performed wind tunnel tests with two identical flags placed side by side and confirmed this destabilisation. In addition, they found that when d is too small (d < 0.2), U_c∗ ac-tually becomes much higher than the critical velocity of one single flag. They argue that the very small d actually makes the two flags to behave as one single flag of a larger thickness, thus a higher flow velocity is required to destabilise the system.

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Still using filaments in soap-film flow, Jia & Yin [Jia 2008] performed experiments with two filaments placed in tandem, i.e. one filament is placed directly down-stream to the other. They investigated flapping patterns and the energy distribu-tion of two filaments by varying the separadistribu-tion distance. Their results show that the downstream filament experiences a drafting induced by the wake of the upstream filament. As a result of the drafting, the drag force on the downstream filament is reduced. Ristroph & Zhang [Ristroph 2008] also conducted experimental work using a soap film and two filaments placed in tandem but found a result opposite to Jia & Yin: the upstream filament actually experiences an inverted drafting. The drag applied on the upstream flag is lower than the drag on the downstream flag. The reason that they found opposite results lies probably in the different leading edge conditions used in these two studies: in [Jia 2008], the upstream filament is fixed at its leading edge, while the leading edge of the downstream filament is tethered by a silk fibre fixed at the other end; in [Ristroph 2008], both flags have a fixed leading edge. The inverted drafting is confirmed by Alben [Alben 2009b] using a vortex sheet model, while Kim et al. [Kim 2010], using an improved version of IBM, reported that both drafting and inverted drafting can be observed depending on the phase difference of both flags’ vortex shedding.

During the last decade, an increasing number of researchers are interested in the coupled dynamics of three or more flexible bodies in uniform flow. Schouveiler et al. [Schouveiler 2009] performed wind tunnel tests with three and four side-by-side flags. Their work reported three possible flapping modes of three flags: (i) in-phase mode, i.e. all three flags have the same vertical motion, (ii) out-of-phase mode, i.e. two consecutive flags have opposite vertical motions, and (iii) symmetrical mode, i.e. the flag in the middle is stretched-straight, while the other two have opposite vertical motions. Michelin & Llewellyn Smith [Michelin 2009] extended the double-wake method to three and more side-by-side flags and investigated their linear stability: for the case of three flags, they also found the three modes reported in [Schouveiler 2009], and for the case of an infinite number of flags, the out-of-phase mode is found to be the dominant one for small M∗ and large d, while for other parameters, the authors reported the existence of modes with the phase difference of any value between 0 and π.

1.2.3 Concluding remarks: why we choose piezoelectric flags

The choice of the flag’s flutter instability as an energy-harvesting mechanism is motivated by its periodic, large-amplitude post-critical motion, which is the main feature of the the flag’s flapping dynamics. Such motion involves a permanent energy exchange between the flags and the surrounding fluid. Many researchers are getting interested in this energy exchange and are seeking ways to harvest energy from it. In general, energy harvesting based on flapping flags may follow two routes: producing energy either from the displacement [Tang 2008,Virot 2015] or from the deformation of the flag [Allen 2001,Singh 2012a]. The latter route has recently been the focus of several studies based on active materials [Doaré 2011a,Dunnmon 2011,

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1.3. A brief introduction to piezoelectricity 9

Giacomello 2011,Akcabay 2012,Michelin 2013]. In our work, we will be interested in piezoelectric materials. The piezoelectric material is chosen in this work for its property of converting a part of mechanical energy generated from mechanical deformation to the electrical energy. In the next section, a brief introduction to piezoelectricity will be presented.

1.3 A brief introduction to piezoelectricity

Piezoelectric materials, as the name indicates, give an “electric” response under “pres-sure”. More precisely, such materials produce electric charge displacement when they are deformed. The discovery of piezoelectricity is attributed to Jacques & Pierre Curie [Curie 1880a, Curie 1880b]. However, their work in 1880 only revealed one piezoelectric effect, the effect that generates electric charge from the material’s defor-mation, since called direct piezoelectric effect. The other effect, called inverse piezoelectric effect remained in shadow at that time until one year later another French physicist, Gabriel Lippmann, who announced that according to the princi-ple of electric charge conservation, a piezoelectric crystal should experience a slight deformation under the influence of an external electric field [Lippmann 1881], a con-clusion that, although based solely on mathematical arguments, was experimentally confirmed in the same year by Jacques & Pierre Curie [Curie 1881].

In 1880, the Curie brothers published their work with the following statement [Curie 1880b]:

Quelle que soit la cause déterminante, toutes les fois qu’un cristal hémiè-dre à faces inclinées, non conducteur, se contracte, il y a formation de pôles électriques dans un certain sens; toutes les fois que ce cristal se dilate, le dégagement d’électricité a lieu en sens contraire.

In this paragraph written in French, the Curie brothers, studying only the quartz, a naturally piezoelectric crystal, reached the conclusion that the reason of electric charge generation under deformation is the formation of electric dipoles within the material itself. This conclusion applies to almost all known piezoelectric materials, though the origin of electrical dipoles varies according to the specific category where a material lies [WEB2 ,Ramadan 2014].

Knowing that electrical dipoles are the origin of piezoelectricity, we are able to describe in a qualitative way how the direct and inverse piezoelectric effects are produced. On one hand, without any deformation, a piezoelectric material is electri-cally neutral, implying that its centres of both positive and negative charges coincide. These two centres would be separated if the material is stretched, compressed, or sheared, creating an electrical field within the material, therefore a voltage difference is generated. An electric charge displacement would occur if the material’s positive side and negative side are connected with a conductive wire, hence the direct piezo-electric effect. On the other hand, an externally applied piezo-electric field would disrupt the electrical neutrality of a piezoelectric material. In order to restore the neutrality,

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O 2-Ti4+_{, Zr}4+ Pb2+ Mechanical strain Induc ed el ec tr ic f iel d External e le ct ric f ie ld Induced strain (a) (b) (c) x y z

Figure 1.7: (a) Elementary cell structure of PZT and piezoelectric effects: (b) di-rect piezoelectric effect and (c) inverse piezoelectric effect (adapted from [WEB2,

Thomas 2011]).

the initially coinciding positive and negative centres would repulse each other as to create an electric field that compensates the external one. As a result, a mechan-ical deformation occurs, and an additional stress is induced due to the material’s elasticity, hence the inverse piezoelectric effect.

Take the elementary crystal structure of PZT (Lead zirconate titanate) for ex-ample. Figure 1.7a shows that when the structure is neutral, its centres of positive and negative charges are at the same point. When a mechanical strain on the x direction is applied on the structure (Fig. 1.7b), these two centres are moved away from each other, in the y direction, thus creating an electric field directing from the positive centre to the negative centre. If an external electric field following the increasing y direction is applied on the structure, in order to restore the electric neu-trality within the structure, the centres of positive and negative charges will move towards opposite directions, inducing consequently a deformation of structure.

Existing piezoelectric materials can fall into four categories: quartz, ceramics, polymers, and composites [Vijaya 2012].

Quartz

Quartz is the crystalline form of silicon dioxide, and it is a naturally occurring piezoelectric material. Besides a strong piezoelectricity, the quartz also possesses other interesting characteristics such as high stiffness, long life, and low sensitivity to temperature and other environmental changes, which makes it an ideal choice for devices requiring a precise frequency control, such as electronic watches, clock signal generators for computers and microprocessor-based instruments. However, its high stiffness and fragility also make it unsuitable for applications involving large deformation. Also, its unique crystalline structure makes it difficult to be shaped

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1.4. Energy harvesting using piezoelectric materials 11

into desired forms.

Ceramics

Piezoelectric ceramics, such as PZT and Barium Titanate, are ferroelectric materials that hold excellent piezoelectric properties. Although quartz has relatively stronger piezoelectricity than ceramics, an advantage of ceramics over quartz is that ceramics are prepared in the form of powder and can therefore be easily shaped into any required geometries, such as discs, cylinders, plates, or thin films, which makes them more widely used for actuator and transducer applications. However, their applications are also limited by their high stiffness and brittleness.

Polymers

The most commonly known and widely used piezoelectric polymer is the Polyvinyli-dene Fluoride (PVDF). Polymers offer rather weak piezoelectricity compared with quartz and ceramics, but have a special advantage that they are flexible and me-chanically more stable. They are usually obtained in the form of large-area thin films which can afterwards be cut into any required dimension. Their excellent flex-ibility makes them a perfect choice for applications involving large deformation and therefore will be used in the present work.

Composite piezoelectric materials

Composite piezoelectric materials are usually made of piezoelectric ceramics, offering strong piezoelectric effects, and softer materials such as polymer or resin that help to improve the material’s flexibility. As a result, composite piezoelectric materials are also interesting candidates when large deformations are required. This material will also be tested in the present work.

1.4 Energy harvesting using piezoelectric materials

Due to their ability of converting the energy associated with mechanical deforma-tion into the electrical energy, piezoelectric materials have been extensively stud-ied in applications of vibration control or suppression [Hagood 1991,Bisegna 2006,

Thomas 2009,Thomas 2011,Ducarne 2012]. In recent years, many publications on energy-harvesting techniques based on piezoelectric materials start to appear, and this section will be dedicated to a brief review of the existing work.

The concept of piezoelectric energy generator emerged around two decades ago [Williams 1996, Umeda 1996]. Its basic idea is to convert ambient vibration en-ergy to useful electrical enen-ergy through piezoelectric materials implemented on vi-bration sources. Many researchers have contributed to this field in order to im-prove the efficiency of such energy-harvesting systems [Sodano 2004, Anton 2007,

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resistive-inductive circuits combined with the piezoelectric material’s intrinsic ca-pacitance [Yang 2005], offer promising opportunities to achieve high efficiency [Shenck 2001,De Marqui 2011].

Flow energy harvesting can be achieved by exploiting the unsteady forcing of the vortex wake generated by an upstream bluff body to force the deformation of a piezoelectric membrane [Allen 2001,Taylor 2001,Pobering 2004]. Fluid-solid insta-bilities offer a promising alternative as they are able to generate spontaneous and self-sustained structural deformation of the piezoelectric structure, e.g. cross-flow instabilities [Kwon 2010,De Marqui 2011,Dias 2013]. In their work, De Marquis et al. [De Marqui 2011] used a resistive circuit and a resistive-inductive one, and found in addition to the beneficial effect of the resonance to the energy harvesting, that a resistive-inductive circuit may also affect the stability of the vibration source. How-ever, the resonant circuit’s influence on the structure’s dynamics was not reported in this work.

Some researchers have already worked on piezoelectric flags [Dunnmon 2011,

Doaré 2011a, Akcabay 2012, Michelin 2013]. In particular, Doaré & Michelin [Doaré 2011a, Michelin 2013] considered a piezoelectric flag coupled with a purely resistive output. They observed moderate efficiency, which is maximised when the characteristic timescale of the circuit is tuned to the frequency of the flag. A signifi-cant impact of the circuit’s properties on the fluid-solid dynamics was also identified.

1.5 Introduction of numerical models used in the present

work

1.5.1 Modelling of the fluid-structure system

In our study, we are interested in the configuration illustrated in Fig. 1.8 (left), where a flag of length L and height H is placed in an uniform flow of velocity U∞.

Figure 1.8: (left) A schematic representation of a flag flapping in an axial flow and (right) the view in x − y plane, where the flag’s motion takes place.

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1.5. Introduction of numerical models used in the present work 13

This representation is again simplified by posing two kinematic assumptions: • the flag’s motion is confined exclusively in the (x, y) plane (Fig. 1.8, right),

i.e. all motions depending on the z direction are neglected;

• the flag is inextensible, i.e. the flag cannot be stretched or shortened.

Under these two assumptions, the flag’s motion consists simply on a bending along the z direction, and consequently the Euler-Bernoulli beam model is used to describe the flag’s dynamics [Antman 1995]. This model is given as follows:

µ∂ 2_x ∂t2 = ∂ ∂s T τ − ∂M ∂s n + Ffluid, (1.3) ∂x ∂s = τ . (1.4)

Equation (1.3) shows that the flag’s displacement x, a function of the Lagrangian coordinate s and time t, is influenced by three actions: the tension T along s, which also ensures the flag’s inextensibility, the bending moment M, which depends on the flag’s bending rigidity B as well as the local curvature, and the fluid forcing Ffluid. With the constitutive law of Euler-Bernoulli beam, the bending moment B

is given by:

M = B∂θ

∂s. (1.5)

Appropriate boundary conditions are required to complete Eq. (1.3). The bound-ary conditions that will be considered in the present work are the so-called clamped-free boundary conditions. These boundary conditions stipulate that the fixed end, or the leading edge of the flag in our work, is “clamped”, meaning that neither dis-placement nor rotation is allowed. At the same time, the free end, or the trailing edge of the flag in our work, is not constrained by any external object apart from the flag itself. This condition entails the cancellation of three quantities at the trail-ing edge: the normal tension T , the bendtrail-ing moment M, and its first derivative in space, representing the shear force on the flag’s cross section.

at s = 0 : x = θ = 0, (1.6)

at s = L : T = M = ∂M

∂s = 0. (1.7)

Equation (1.4) represents the inextensibility condition, which implies that the tension T is computed, using the boundary condition given by Eq. (1.7), by inte-grating the tangential component of Eq. (1.3) from s = L:

T (s, t) = Z s L µ∂ 2_x ∂t2 · τ − B ∂θ ∂s ∂2θ ∂s2 − Ffluid· τ ds0 (1.8)

The previously mentioned LABET is chosen to model the fluid loading Ffluid. In

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order expression for the “reactive force” F_react which is computed from the relative velocity of the flag to the incoming flow:

Unn + Uττ =

∂X

∂t − U∞ex. (1.9)

The reactive force can then be derived by computing the advection of the fluid added momentum by the flow, which is an inviscid effect. Here, instead of presenting the rigorous proof of the LAEBT [Lighthill 1960, Lighthill 1970, Lighthill 1971], we only show a simplified derivation of the reactive force expression in terms of momentum conservation.

Assuming that the flag’s motion is confined in the x − y plane, we consider two planes Π(s) and Π(s + ds), both perpendicular to the x − y plane, intersecting the flag at s and s + ds, thus defining a segment of length ds on the flag along its streamwise direction. We then compute the momentum variation of the fluid flow passing across the zone confined by the planes Π(s) and Π(s + ds) (See Fig.1.9for notations).

Figure 1.9: Configuration used for reactive force derivation.

Lighthill pointed out that the essential characteristic of an “elongated body” is that the fluid added mass with respect of the relative motion in the normal direction n is large, whereas in the tangential direction τ , the added mass is small [Lighthill 1971]. As a result, the reactive force is principally due to the variation of the normal component of the fluid’s added momentum p_n in the volume defined by I + II. The expression of pn is given by:

pn(s) = MaUn(s)n(s)ds, (1.10)

where Ma = maρfH2 is the added mass of the fluid surrounding the flag, and we

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width H [Lighthill 1971]. The local conservation of p_n(s) in the volume I + II is given by the following expression:

∂ ∂t(MaUnn) = ∂ ∂s(MaUnUτn) − 1 2 ∂ ∂s(MaU 2 nτ ) − Freact. (1.11)

The three terms on the right-hand side of Eq. (1.11) represent three contributions to the normal momentum variation, which are:

• the advection by the flow along the tangential direction τ ;

• the difference of the fluid pressure acting on Π(s) and Π(s + ds) along the tangential direction τ , which, according to [Lighthill 1970] can be obtained using Bernoulli’s theorem;

• the flag’s force as a reaction to the fluid pressure, which is the opposite of the reactive force, thus noted as −F_react.

Using the following relations: ∂n ∂t = − ∂θ ∂tτ , ∂n ∂s = − ∂θ ∂sτ , ∂τ ∂s = ∂θ ∂sn, (1.12)

we can rewrite Eq. (1.11) into the following form: Freact= −Ma ∂Un ∂t − ∂ ∂s(UnUτ) + 1 2U 2 n ∂θ ∂s n + Un ∂θ ∂t − Uτ ∂θ ∂s − ∂Un ∂s τ . (1.13) By calculating the time derivative of Eq. (1.9), and using the inextensibility condi-tion of Eq. (1.4), we may obtain the following result:

∂θ ∂t − Uτ ∂θ ∂s − ∂Un ∂s = 0, (1.14)

thus cancelling the tangential component in Eq. (1.13). Replacing Ma by maρfH2

in Eq. 1.13, we obtain therefore the expression of the reactive force: Freact= −maρfH2 ∂Un ∂t − ∂ ∂s(UnUτ) + 1 2U 2 n ∂θ ∂s n. (1.15)

Candelier et al. [Candelier 2011] recently proposed an analytic proof of this result, and successfully compared it to RANS simulations for fish locomotion problems. These authors also stated that in the case of spontaneous flapping, it is necessary to account for the effect of lateral flow separation, which is a 3D effect. Here, we choose an empirical “resistive” force model given by the following term [Taylor 1952,

Eloy 2012,Singh 2012b]:

Fresist = −

1

2ρHCd|Un|Unn, (1.16)

where Cd = 1.8 is the drag coefficient for a rectangular plate in transverse flow

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Figure 1.10: (a) Schematic representation of a flag covered by straps of piezoelectric materials and (b) schematic view of pairs of piezoelectric patches.

terms that account for the skin friction, which is irrelevant to the present work. The fluid forcing is the sum of these two terms:

Ffluid= Freact+ Fresist. (1.17)

The applicability of this result to flapping flag was confirmed experimentally, at least up to an aspect ratio H∗ = 0.5, the value considered in our work [Eloy 2012].

We may notice that the viscous effect is neglected in the above model. This assumption is justified by the fact that the flapping flag dynamics considered here implies a large Reynolds number. A 10 cm long/wide flag in a wind flowing at around 5 m/s, or a water current of around 0.5 m/s leads to Re ∼ 104. Such order of magnitude of Re can also be obtained using configurations of existing experimental works [Eloy 2012,Virot 2013].

1.5.2 Piezoelectric effects

The surface of the flag is covered by piezoelectric patches (Fig.1.10a). These patches are placed by pairs, i.e. every patch on one surface is paired with an identical one on the opposite surface. The left and right ends of the ith pair are respectively denoted by s−_i and s+_i (Fig. 1.10b). When deformed, each patch generates an electric charge displacement Qk_i due to the direct piezoelectric effect. This quantity is given by the following equation [Ducarne 2012]:

Q(k)_i = χ[θ]s + i s−_i + C (k)_V(k) i , (1.18)

where χ is a mechanical/electrical conversion factor quantifying the proportion of the flag’s deformation energy being converted to the electrical energy. V_i(k) is the voltage difference between the two electrodes of the ithpiezoelectric patch. And C_i(k) is the intrinsic capacity of the ithpiezoelectric patch [Erturk 2009]. Equation (1.18) applies to both patches in the ithpiezoelectric pair. The superscript k takes values 1

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or 2, indicating one surface and the other of the flag (Fig.1.10b). If the two patches are connected in series in the same loop, with reversed polarity, the consequence is that since each patch experiences a deformation opposite to the other, the same quantity of electric charge passes through these two patches in the same direction, and that the same voltage is applied on each patch. If Qi and Vi denote respectively

the electric charge and the voltage of the ith piezoelectric pair, we obtain following expressions:

Qi = Q(1)i = Q (2)

i (1.19)

Vi = 2V_i(1)= 2V_i(2) (1.20)

Also, since two patches are identical, they have the same intrinsic capacity. The equivalent capacity of the ithpair, as a result of a connection in series, is half of the intrinsic capacity of each patch:

Ci =

C_i(1)

2 =

C_i(2)

2 (1.21)

The inverse piezoelectric effect, i.e. the elastic stress induced by an external voltage, is modelled as an additional torque applied on the segments of the flag that are covered by piezoelectric pairs [Thomas 2009]:

Mpiezo= −

X

i

χViFi (1.22)

where the function F is defined as: Fi(s) =

(

1 if s−_i < s < s+_i

0 if elsewhere (1.23)

Piezoelectric effect therefore adds a new contribution on the bending moment ap-plied on the flag, which is now written as:

M = B∂θ ∂s −

X

i

χViFi. (1.24)

Finally, a relation exists between the electric charge Q_iand the voltage V_i, which depends on our choice of external circuits. This relation will for now be written in its generic form as:

F (Vi, Qi) = 0 (1.25)

The complete system of equations describing a flag covered by a finite number of piezoelectric patches is therefore given by:

µ∂ 2_x ∂t2 = ∂ ∂s " T τ − ∂ ∂s B ∂θ ∂s − X i χViFi ! n # + Ffluid, (1.26) Qi = χ[θ] s+_i s−_i + CiVi, (1.27) F (Vi, Qi) = 0. (1.28)

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And the clamped-free boundary conditions are given by: at s = 0 : x = θ = 0, (1.29) at s = L : T = B∂θ ∂s − χV = B ∂2θ ∂s2 = 0. (1.30) 1.5.3 Dimensionless equations

We suppose first that all piezoelectric pairs are of identical length, l_p, and introduce the linear density of capacitance c, defined as:

c = C lp

. (1.31)

Equations (1.26)–(1.30) are then written into dimensionless form using L, L/U∞

and ρ_fHL as characteristic scales of length, time and mass, respectively. We choose U∞

√

µc as the characteristic scale for linear density of electric charge (Qi/lp), and

U∞pµ/c for voltage (V ). We naturally obtained three dimensionless parameters

M∗, U∗ and H∗ defined previously. An additional dimensionless parameter, α is found to characterise the piezoelectric coupling:

α = √χ

Bc. (1.32)

This parameter measures the intensity of the mutual forcing between the piezoelec-tric patches and the flapping flag, therefore would command a crucial impact on the energy transfer between the flag and the patches, and consequently on the en-ergy harvesting performance. We also expect that other dimensionless parameters, characterising electrical properties of the system, will be obtained according to the nature of the circuit to be used.

The flag, the piezoelectric patches, and an electric circuit, constitute the coupled system at the core of the present work. The behaviour of this system is therefore described by the following system of dimensionless equations:

∂2x ∂t2 = ∂ ∂s " T τ − ∂ ∂s 1 U∗2 ∂θ ∂s − 1 U∗ X 1 αViFi ! n # + M∗Ffluid, (1.33) Qi = α U∗ [θ] s+_i s−_i + Vi, (1.34) F (Vi, Qi) = 0. (1.35)

And the dimensionless boundary conditions are:

at s = 0 : x = θ = 0 (1.36) at s = 1 : T = 1 U∗2 ∂θ ∂s − α U∗V = 1 U∗2 ∂2θ ∂s2 = 0. (1.37)

The above system of equations serves as the root of the following chapters of this work. In addition, two different models will be used to compute the fluid forcing:

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1.6. Energy harvesting 19

Figure 1.11: Rectangular section defined by amplitude A and width H. one model is the LAEBT presented previously of which the expression is given by Eqs. (1.15)–(1.17), and the other model is the vortex sheet model introduced in [Alben 2009a].

In Eqs. (1.33)–(1.34), same notations are used for dimensionless variables x, s, t, V and Q. Throughout this manuscript, no distinction will be made between dimensional and dimensionless variables.

1.6 Energy harvesting

The last important part of the modelling stage is to define the harvested energy. As such definition cannot be based on a specific everyday application (e.g a fan, a lamp bulb, or a mobile phone charger...) at the current stage, we will simply consider the most common way of electrical energy consumption: the Joule effect. More precisely, a resistor is placed in the circuit and the dissipation in this resistor is defined as the harvested energy. Although an expression of the harvested power, noted by P hereinafter, depends on the configuration of a given circuit and therefore is not given at this stage, a definition of the efficiency can already be obtained by choosing a reference. As in other studies on energy harvesting involving flags, we choose the kinetic energy flux of the fluid flow passing through the rectangular section delimited by the flag’s width and its peak-to-peak amplitude 2A (Fig.1.11).

The efficiency η is therefore given by the following expression: η = ₁ < P >

2ρfHLU∞3 × 2 < A >

, (1.38)

where A is the amplitude of the flag and the operator <> gives the time average of a quantity. By using the quantity ρ_fL2HU_∞2 as the characteristic scale of energy, the efficiency is given in non-dimensional form by:

η = < P >

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where A now denotes the dimensionless amplitude of the flag.

1.7 Energy harvesting using piezoelectric flag connected

to resistive circuits

We choose to single out this part as we consider the studies using resistive circuits constitute an important precursor of our work. Some previous works [Dunnmon 2011,Doaré 2011a,Michelin 2013] reported results obtained using a cir-cuit containing only a resistor. The generic relation between electric charge and voltage given by Eq. (1.25), for this particular circuit becomes:

V + R∂Q

∂t = 0, (1.40)

where R represents the resistor connected in the circuit.

In [Dunnmon 2011], the authors reported an experimental study using piezoelec-tric patch based on PZT. They argued that a considerable amount of energy would be harvested using piezoelectric materials if aeroelastic energy harvesters could be carefully designed as to optimise relevant operating parameters.

Such optimisation is reported in [Doaré 2011a,Michelin 2013] where the authors used theoretical and numerical tools to study both a piezoelectric flag’s linear sta-bility [Doaré 2011a] and post-critical dynamics [Michelin 2013] when it is connected to resistive circuits. They pointed out that:

• in terms of the linear stability [Doaré 2011a], the resistance stabilises the sys-tem when the mass ratio M∗ is small, while a destabilising effect of resistance is observed with large M∗, consistent with previously reported results on the destabilisation by damping [Doaré 2010];

• a tuning between the circuit’s characteristic time scale and the flapping pe-riod leads to a maximal efficiency. Within the range of parameters used in [Michelin 2013], a maximal efficiency of around 10% is achieved with M∗ ∼ 20. The results also suggest that higher efficiency can be expected with even larger M∗.;

• a critical impact on the coupled system’s dynamics by the piezoelectric cou-pling is observed [Doaré 2011a, Michelin 2013]. The authors reported that when the piezoelectric coupling, characterised by α, is strong, both the flap-ping frequency and amplitude showed modifications, resulting from the damp-ing induced by resistor.

The results in [Doaré 2011a,Michelin 2013] highlight for the first time the im-portant role of piezoelectric coupling in the energy harvesting: the efficiency is found to be scaled as α2. The authors also found that the choice of electric circuits for energy harvesting might have a critical impact on the system’s dynamics and per-formance, even when the most elementary local resistive circuit is concerned. It

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1.8. Outline of manuscript 21

is therefore reasonable and interesting to expect that other kinds of circuits would bring about different impacts to the system, influencing consequently the energy-harvesting efficiency. Studying impacts of different circuits constitutes therefore an important part of the present work.

1.8 Outline of manuscript

The present work will be elaborated in four chapters. Chapters 2–4 will focus on impacts of output circuits on the energy harvesting as well as on the dynamics of the flags. In Chapter2, we will study a flag totally covered by one pair of piezoelectric patches, and connected to a resistor as well as an inductor, which, combined with the piezoelectric pair’s intrinsic capacity, provides an additional natural frequency to the system. We will study, principally by experimental means but also numerically, the resonant property of the system and its impacts on the energy harvesting per-formance. In Chapter3, the other case of piezoelectric coverage will be investigated: we will consider that the flag is totally covered by pairs of piezoelectric patches of in-finitesimal size. This type of coverage can strengthen the inverse piezoelectric effect on the flag’s dynamics, as the additional torque is adapted to the local curvature of the flag. Each piezoelectric pair on the flag is connected to a resistive-inductive circuit. In this chapter, we identify a strong coupling phenomenon between the piezoelectric flag and the circuit and the energy-harvesting performance is consid-erably improved compared with results reported in [Michelin 2013]. In Chapter 4, instead of connecting each piezoelectric pair to separated circuits, we will use an electric network that interlinks these piezoelectric pairs. The impacts on the energy harvesting as well as the flag’s dynamics of the electric network will be studied in this chapter. In Chapter 5, we will turn to the coupled flutter of two flags, and the way it impacts the energy harvesting performance.

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Chapter 2

Single Piezoelectric Coverage

In this chapter, a flag covered by one single pair of piezoelectric patches will be studied. As a natural continuation of previous works on purely resistive circuits [Dunnmon 2011,Doaré 2011a,Michelin 2013], a different circuit will be used. This new circuit contains two components: a resistor, which is indispensable for modelling the energy harvesting, and an inductor, which introduces resonant properties to the circuit. We will mainly focus on experimental studies. Numerical studies will also be carried out to complement the experimental results. In addition, a simple current source model will be used to conjecture whether piezoelectric effects apply an impact on the flapping dynamics.

2.1 Experimental set-up

Our experiments are conducted using two prototypes of piezoelectric flag made of two different piezoelectric materials: Polyvinylidene Difluoride (PVDF) and Macro-Fibre Composite (MFC). To fabricate these prototypes, two piezoelectric patches, of identical size, are glued face-to-face and connected with reversed polarity, forming therefore a flag covered by one single piezoelectric pair.

Experiments with these two flags are performed in a same wind tunnel, here-inafter referred to as Tunnel A, whose test section is 10 cm’s wide and 4 cm’s high [Doaré 2011c] (Fig. 2.1a). The maximal flow velocity that Tunnel A could reach is around U∞ = 50 m/s. The wind tunnel’s shape therefore prescribes the limits of

our prototypes’ dimension, as well as the way they are placed in the wind tunnel. PVDF being a paper-like, soft and easily sizeable polymer, the flag based on this material is made to be 9.5 cm’s long and 2.5 cm’s wide so that it can be placed in Tunnel A in the way shown in Fig. 2.1a. This positioning of flag minimises the influence of the flag’s weight on its flapping motion as the gravity acts in the span-wise direction, which is perpendicular to the transverse direction. MFC, however, is a prefabricated patch which cannot be reshaped. As the patch’s dimension is 9.3 cm×8.9 cm, prohibiting a similar positioning as the PVDF flag, it has to be placed horizontally in Tunnel A (Fig. 2.1b). With this positioning, the gravity acts in the transverse direction, which is the same direction as the flag’s flapping motion. It is therefore necessary to evaluate the importance of the gravity’s influence on the flag’s dynamics. One can compare the order of magnitude of the gravity-induced torque on the flag, with that of the elastic torque of the flag:

ε = O gravity-induced torque elastic torque = O mgL 2 EHh3 . (2.1)

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Figure 2.1: Experimental setup: (a) a PVDF piezoelectric flag and (b) a MFC piezoelectric flag in Tunnel A. Both flags are connected with an external circuit and measurement device (c).

Length Width Thickness Mass Young’s Modulus PVDF 9.5 cm 2.5 cm 40 µm 5.5×10−4 kg 2 GPa MFC 9.3 cm 8.9 cm 0.4 mm 1.3×10−2 kg 20 GPa Table 2.1: Geometric and mechanical properties of piezoelectric patches.

In Eq. (2.1), m is the mass of one MFC patch, E is the MFC’s Young’s modulus, and L, H, h are respectively the length, the width and the thickness of the MFC patch. These quantities are shown in Table 2.1. One can obtain that for MFC, ε ∼ 10−2. This result shows that compared with the MFC flag’s rigidity, the gravity’s effect is relatively small and can be neglected. As a result, for any tests performed on the MFC flag positioned as in Fig. 2.1b, one can assume that the gravity is absent. Visually, it is also observed that the MFC flag remains horizontal in its streamwise direction when the flow velocity is 0, which confirms our assumption.

Evidently, for each individual test, only one flag is placed in the wind tunnel. The first series of experiments are performed with the PVDF flag which is clamped to keep a length of 8.5 cm, and the MFC flag. Their respective motion are shown in Fig. 2.2.

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2.2. Comparison between PVDF and MFC 25

Figure 2.2: Flapping of (left) a PVDF flag and (right) an MFC flag

Figure 2.3: “Effective length” of the PVDF flag.

resistor ranging from 5 Ω to 108 Ω. This circuit is then connected to the flag, and to a data acquisition board (DAQ) (Fig.2.1c), which records the output voltage V using the software LabView R_{. Based on V , one may compute the harvested power}

P:

P = V

2

R. (2.2)

2.2 Comparison between PVDF and MFC

Based on the descriptions in the previous section, one can infer that the MFC flag is more rigid then the PVDF flag, and will consequently wonder whether within the flow velocity range of the wind tunnel, both flags would become unstable and flap. We start by examining the critical velocity for the PVDF flag of different values of L. As the total length of the PVDF flag is 9.5 cm (Table 2.1), we vary the

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0.04

5

0.05

0.06

0.07

0.08

10

15

20

25 L

_(m)

U

_c

_(m/s)

U

_cu

U

_cd

Figure 2.4: Critical instability velocity for the PVDF flag in Tunnel A.

“effective length”, i.e. the length of the flag’s portion exposed to the fluid flow and is susceptible to flap, by adjusting the position at which the flag is clamped. The other portion of the flag is attached to a rigid plate parallel to the flow so that its influence on the flow would be minimised (Fig. 2.3). For each L, we first increase the flow velocity U∞ from 0 to a value Ucu at which the flag starts to flap; then the

flow velocity is decreased from U_cd to another value U_cu at which the flapping stops. In Fig.2.4, the values of U_cu and U_cdobtained using the PVDF flag are plotted with different L.

We observe first that the critical velocity U_cshown in Fig.2.4is consistent with existing work on experimental study in two ways:

• Many works studying the critical velocity have reported a bistability [Alben 2008a, Michelin 2008, Eloy 2008, Eloy 2012, Virot 2013]: the coexis-tence of flapping and motionless states over a certain range of flow velocity. This phenomenon is also observed in the present experiment using PVDF flag. For every value of L, we observe the existence of bistability over a flow velocity range, of which the lower boundary is always U_cd, and the higher boundary is U_cu.

• For a given value of width H, longer flags become unstable at a lower velocity. This observation is consistent with experimental results reported in previous work [Yamaguchi 2000b,Yamaguchi 2000a,Eloy 2007,Virot 2013].

Moreover, the measurement of critical velocity of the PVDF flag shows that the flag becomes unstable at reasonable flow velocities, well below the Tunnel A’s limitation. We can therefore expect to perform tests using the PVDF flag with a large range of U∞and L.

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2.3. Modelling of a flag covered by one piezoelectric pair 27

Figure 2.5: Equivalent circuit of experimental set-up.

The MFC flag is however more difficult to be set in motion. As MFC is a composite based on PZT, a piezoelectric ceramic, it inherits the rigidity of PZT. A considerably large flow velocity is mandatory to destabilise the flag. To accelerate the flow in Tunnel A beyond its limitation, the wind tunnel’s cross section has to be further reduced, leaving a height of only 2.5 cm, which limits significantly the flag’s flapping amplitude. Using the MFC flag of the dimension indicated in Table 2.1, The measurement gives U_cu = 57.6 m/s, a value close to the maximal velocity that Tunnel A is capable of achieving even under further confinement, and U_cd = 49.5 m/s, indicating the existence of the bistability. The results of critical velocity of MFC flag exclude the idea of reducing further its “effective length”, which would surely lead to another critical velocity that is prohibitively large for Tunnel A.

From the above comparison, one may conclude that the PVDF flag offers a wider manoeuvrability for experimental study: within the limitation of experimental devices, we can perform tests over a larger range of U∞ and L, while with the MFC

flag, only one value of L and a very small range of U∞are available. For this reason,

this chapter will mainly focus on studying the PVDF flag, using both experimental and numerical means. Some experimental results obtained using MFC flag will only be briefly presented at the chapter’s end.

2.3 Modelling of a flag covered by one piezoelectric pair

2.3.1 Simple current source model neglecting piezoelectric feed-back

The system composed of the flag, the resistive-inductive circuit, and the DAQ can be described using the equivalent circuit shown in Fig.2.5. According to Fig. 2.5, the equivalent circuit consists of an electric current source in parallel connection with an inductor of inductance L, a resistor of resistance R, and a capacitor C which comes from the intrinsic capacitance of piezoelectric patches. For the sake of consistency in notation, in the following text, we will use C = cL, where c is the linear density of capacitance, as introduced in Chapter1, and L is the effective length of the flag,

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as shown in Fig. 2.3. This equivalent circuit also takes into account the internal resistance of the inductor RL, which is connected in series with the inductance, and

that of the DAQ, Rd, which is in parallel connection with other elements.

From the voltage signal as shown in Fig. 2.6, we can assume that the current source’s output takes a harmonic form:

I(t) = I0ejωt, (2.3)

where ω = 2πf with f being the flapping frequency. The amplitude I₀ depends on the conversion factor χ and the flag’s leading edge angle while flapping. Using this model, we can compute analytically the dissipation rate in R, i.e. the harvested power P:

P = IRIR∗R, (2.4)

where I_R represents the current passing through R, given by:

IR= jωI0ejωt 1 + R jLω + RL + R Rd + jωCR , (2.5)

and I_R∗ is its complex conjugate.

While this model does not replace the nonlinear numerical simulation, we use it as a mean of conjecturing whether an inverse effect is applied on the flag’s dynamics, as no such effect is included in this model. In the upcoming experimental study, the prediction obtained from this model, referred as Simple Current Source Model hereinafter, will be compared against experimental results. The conjecture is based on the fact that flag’s dynamics would remain unaltered under weak inverse piezo-electric effect, therefore good agreement would be found between experimental data and Eq. (2.4), whether or not the system is at resonance. If the inverse piezoelectric effect is strong and that the dynamics of the flag is impacted, the Simple Current Source Model would no more agree with experiments. In the Simple Current Source Model, we choose a constant amplitude by supposing that a permanent flapping regime is reached, and that the coupling is weak enough so that the flag’s dynamics is not influenced by the inverse piezoelectric effect. Theoretically, this amplitude depends on both the angle of the trailing edge and the mechanical/electrical con-version factor χ. In our work, I0 is determined using a purely resistive circuit: first,

experiments are performed using a purely resistive circuit with varying R, and then I0 is chosen as the value which allows the Simple Current Source Model, also using

a resistive circuit, to show a good agreement with experimental data.

2.3.2 Nonlinear numerical model

As the flag is covered by one single pair of piezoelectric patches, the expressions of piezoelectric effects, derived from Eqs. (1.18) and (1.22) are written as:

Q = χθ(s = L) + clV, (2.6)

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2.3. Modelling of a flag covered by one piezoelectric pair 29

Equation (2.6) indicates that with a flag totally covered by one single piezoelectric pair, the critical variable relating to the electric charge displacement is the angle of the flag’s trailing edge.

We need to find the relation between Q and V to close our modelling. For the circuit presented in Fig. 2.5, this relation is written as follows:

V + Z ·∂Q

∂t = 0, (2.8)

where Z is the operator of impedance corresponding to the resistor, inductor and the internal resistance of the DAQ in parallel connection. This operator is given as:

Z = 1 1 R+ 1 L∂ ∂t + RL + 1 Rd . (2.9)

In Eq. (2.8), we replace Z by Eq. (2.9), and Q by Eq. (2.6) to obtain the following relation describing the circuit’s behaviour under the forcing of the piezoelectric flag:

ReLcL ∂2V ∂t2 + (ReRLcL + L) ∂V ∂t + (Re+ RL) V + χReL ∂2θ ∂t2 + χReRL ∂θ ∂t = 0, (2.10) where R_e is the equivalent resistance of R and R_d in parallel connection:

R_e = RRd

R + R_d. (2.11)

Using the same characteristic scales introduced in Chapter 1, the non-dimensional form of Eqs. (2.6)–(2.10) is written as follows:

Q = α U∗θ(s = 1) + v, (2.12) M_piezo= − α U∗V, (2.13) ∂2V ∂t2 + βLω02+ 1 βe ∂V ∂t + 1 +βL βe ω₀2V + α U∗ ∂2θ ∂t2 + α U∗βLω 2 0 ∂θ ∂t = 0. (2.14) Two new dimensionless parameters, β and ω0, appear in the above equations.

Their definitions are given below:

β = cRU∞, ω0 = 1 U∞ r L Lc (2.15)

The parameter β characterises the rate of dissipation in a given resistor, there-fore, while β represents the harvesting resistor, the internal resistance of inductor RL and DAQ Rd are represented by βL and βd, respectively. βe represents the

equivalent resistance given by Eq. (2.11). The parameter ω0 is the dimensionless

natural frequency of the circuit.

Equations (2.12)–(2.14) are then incorporated into the system of equations (1.33)–(1.34) to establish the system to be solved for the numerical study.

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2.4 Characterisation of the coupling coefficient α

As defined by Eq. (1.32) in Chapter 1, the coupling coefficient α is determined by three quantities: the conversion factor χ, the bending rigidity B, and the capacitance per unit length c. Among these quantities, c can be measured directly. We first measure C using a multimeter, which gives C = 14 nF. This measured C is then divided by the effective length of the flag to obtain c. Note that since the whole PVDF flag, i.e. both the clamped part and the effective part (Fig.2.3), is connected in the circuit, the total intrinsic capacitance C is the same regardless of the value of the effective length L. As a result, the linear density of capacitance c based on c = C/L depends on the effective length L: larger effective length L leads to lower c.

Some other measurements are required to determine B and χ. In this section, we will simply outline the methodology employed to determine each quantity. More experimental details are to be found in Appendix A.

2.4.1 Measurement of B

As the flag is considered as a three-layer sandwich plate, whose width is H, its bending rigidity B could be computed using a dedicated formula [Lee 1989]:

B = E0h 3 0H 12(1 − ν₀2) + 2EphpH 1 − ν2 p h2₀ 4 + h0hp 2 + h2_p 3 ! , (2.16)

where E are ν are respectively the Young’s modulus and the Poisson’s coefficient of corresponding material, h is the thickness of a layer. The subscripts 0 and p indicate the middle layer and the piezoelectric layers of the flag. However, the middle layer of the flag is a double-sided bonding tape, of which the Young’s modulus E0 and

thickness h0 are difficult to ascertain. It is therefore more practical to measure the

flag’s free vibration frequency f₀, which, at the first vibration mode, is given by [Timoshenko 1953] f0 = 3.515 2πL2 s B µ, (2.17)

where µ is the flag’s mass per unit streamwise length. The bending rigidity is therefore computed from Eq. (2.17).

2.4.2 Measurement of χ

The conversion factor χ is formally given by: χ = e31(hp+ h0)H

2 , (2.18)

where e₃₁ is a piezoelectric coefficient. Again, h₀ is difficult to measure due to the nature of the middle layer made of double-sided bounding tape. Our method consists therefore of exploiting the direct piezoelectric effect given by Eq. (2.6). We

Energy harvesting by piezoelectric flags

HAL Id: tel-01571574

https://tel.archives-ouvertes.fr/tel-01571574

Yifan Xia

To cite this version:

Docteur de l’Ecole Polytechnique

Specialité : Mécanique

Yifan Xia

Energy harvesting by piezoelectric flags

∗

Remerciements

Contents

Chapter 1

Introduction

1.1

Overview of flow energy harvesting

1.2

Flutter instability

(a)

(b)

1.3

A brief introduction to piezoelectricity

1.4

Energy harvesting using piezoelectric materials

1.5

Introduction of numerical models used in the present

work

1.6

Energy harvesting

1.7

Energy harvesting using piezoelectric flag connected

to resistive circuits

1.8

Outline of manuscript

Chapter 2

Single Piezoelectric Coverage

2.1

Experimental set-up

2.2

Comparison between PVDF and MFC

0.04

5

0.05

0.06

0.07

0.08

10

15

20

25

L

(m)

U

(m/s)

U

U

2.3

Modelling of a flag covered by one piezoelectric pair

2.4

Characterisation of the coupling coefficient α

_(m)

_(m/s)